Problem: $ F = \left[\begin{array}{rrr}4 & 4 & 0 \\ -2 & -2 & 4 \\ 3 & 1 & -1\end{array}\right]$ $ D = \left[\begin{array}{r}0 \\ 0 \\ 4\end{array}\right]$ Is $ F+ D$ defined?
Solution: In order for addition of two matrices to be defined, the matrices must have the same dimensions. If $ F$ is of dimension $( m \times  n)$ and $ D$ is of dimension $( p \times  q)$ , then for their sum to be defined: 1. $ m$ (number of rows in $ F$ ) must equal $ p$ (number of rows in $ D$ ) and 2. $ n$ (number of columns in $ F$ ) must equal $ q$ (number of columns in $ D$ Do $ F$ and $ D$ have the same number of rows? Yes Yes No Yes Do $ F$ and $ D$ have the same number of columns? No Yes No No Since $ F$ has different dimensions $(3\times3)$ from $ D$ $(3\times1)$, $ F+ D$ is not defined.